A145518 Triangle read by rows: T1[n,k;x] := Sum_{partitions with k parts p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, for x_i = A000040(i).

2, 3, 4, 5, 6, 8, 7, 19, 12, 16, 11, 29, 38, 24, 32, 13, 68, 85, 76, 48, 64, 17, 94, 181, 170, 152, 96, 128, 19, 177, 326, 443, 340, 304, 192, 256, 23, 231, 683, 787, 886, 680, 608, 384, 512, 29, 400, 1066, 1780, 1817, 1772, 1360, 1216, 768, 1024, 31, 484, 1899, 3119

Offset: 1

Author

Tilman Neumann, Oct 12 2008

Keywords

• nonn
• tabl

Comments

Let p(n; m_1, m_2, m_3, ..., m_n) denote a partition of integer n in exponential representation, i.e., the m_i are the counts of parts i and satisfy 1*m_1 + 2*m_2 + 3*m_3 + ... + n*m_n = n.

Let p(n, k; m_1, m_2, m_3, ..., m_n) be the partitions of n into exactly k parts; these are further constrained by m_1 + m_2 + m_3 + ... + m_n = k.

Then the triangle is given by T1[n,k;x] := Sum_{all p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, where x_i is the i-th prime number (A000040).

2nd column (4, 6, 19, 29, 68, 94, 177, ...) is A024697.

Row sums give A145519.

Examples

			Triangle starts:
   2;
   3,   4;
   5,   6,   8;
   7,  19,  12,  16;
  11,  29,  38,  24,  32;
  13,  68,  85,  76,  48,  64;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Tilman Neumann, More terms, partition generator and transform implementation.

Crossrefs

Cf. A000040, A024697, A145519, A145520, A258323.

Programs

Maple

g:= proc(n, i) option remember; `if`(n=0 or i=1, (2*x)^n,
      expand(add(g(n-i*j, i-1)*(ithprime(i)*x)^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(g(n$2)):
seq(T(n), n=1..12);  # Alois P. Heinz, May 25 2015

Mathematica

g[n_, i_] := g[n, i] = If[n==0 || i==1, (2 x)^n, Expand[Sum[g[n-i*j, i-1]*(Prime[i]*x)^j, {j, 0, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][g[n, n]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

Extensions

Reference to more terms etc. changed to make it version independent by Tilman Neumann, Sep 02 2009